Question: Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$.
Solution: To start, we can find the $y$-intercept of each of these lines.  Using this, we can calculate the length of that side of the triangle, and use it as a base.  Letting $x=0$ in the first equation gives $y=-2$ as a $y$-intercept.  Letting $x=0$ in the second equation gives $3y=12\Rightarrow y=4$ as a $y$-intercept.  Therefore, the triangle has a length of $4-(-2)=6$ on the $y$-axis.

The height of the triangle will be equal to the $x$-coordinate of the intersection of the two lines.  So, we need to solve for $x$ in the system: \begin{align*}
y-3x&=-2\\
3y+x&=12
\end{align*}Multiply the first equation by 3, then subtract the second equation as shown: \begin{tabular}{ r c c c l}
$3y$&-&$9x$&=&-6\\
-($3y$&+&$x$&=&12)\\ \hline
&-&$10x$&=&-18\\
\end{tabular}Therefore, $x=\frac{18}{10}=\frac{9}{5}$.  This is equal to the height of the triangle.  The area will be $\frac{1}{2}\cdot \frac{9}{5}\cdot 6=\boxed{\frac{27}{5}}$